subroutine rgm (n, x, y, output, ier)
 
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    Package:  SLRPACK
 
    Version:  October, 1985
 
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    PURPOSE
    -------
 
    This subroutine computes estimates of simple linear regression
    parameters for geometric mean regression.
 
    DESCRIPTION
    -----------
 
    1.  The input data are observations (x(i), y(i)), i = 1,...,n.
 
    2.  This technique is appropriate for fitting a straight line
 
                           Y  =  A  +  B * X
 
        to observations  (x(i),y(i)),i=1,...,n  where
 
            x(i) = X(i) + e(i)   and   y(i) = Y(i) + d(i),
 
            X(i) and Y(i) are unknown population means, and
 
            e(i) and d(i) are random error terms distributed normally
            with zero mean and variances s**2 = s(e)**2 = s(d)**2.
 
    3.  The name geometric mean alludes to the fact that if B(OLS-y)
        denotes the estimated slope from conventional least squares
        regression (with the y-observations but not the x-observations
        subject to error) and if B(OLS-x) denotes the converse, then
        the geometric mean slope estimate is the geometric mean of
        B(OLS-y) and B(OLS-x).  This technique for estimating the
        regression parameters is preferable to OLS-y and to OLS-x
        for the above type of data because the OLS-y technique is
        known to give under-estimates of the magnitude of the true
        slope and over-estimates of the magnitude of the true intercept
        while OLS-x results have the opposite characteristics (Riggs
        et al (1978)).
 
    4.  This subroutine straightforwardly implements the calculations
        described in the first reference.  The regression line is
        given by equation (2), the standard deviations of the
        estimated slope is given by equation (11), and the standard
        deviation of the estimated intercept is given by equation (14)
        all in the first reference.  The correlation coefficient is
        the Pearson product-moment correlation coefficient.
 
    INPUT PARAMETERS
    ----------------
 
    N      Integer scalar (unchanged on output)
           Number of observations.
 
           N must be greater than 2.
 
    X      Real vector dimensioned at least N (unchanged on output)
           X-observations.
 
    Y      Real vector dimensioned at least N (unchanged on output)
           Y-observations.
 
    OUTPUT PARAMETERS
    -----------------
 
    OUTPUT Real vector dimensioned at least 7 -- output
 
           OUTPUT(1) = Slope of geometric mean regression line
 
           OUTPUT(2) = y-intercept of geometric mean regression line
 
           OUTPUT(3) = Standard deviation of the slope
 
           OUTPUT(4) = Standard deviation of the intercept
 
           OUTPUT(5) = Average of the x-observations
 
           OUTPUT(6) = Average of the y-observations
 
           OUTPUT(7) = Pearson product-moment correlation coefficient
 
    IER    Integer scalar -- output
           Execution error indicator.
 
           IER = 0   No errors
 
           IER = 1   Cannot compute parameters by geometric mean technique
                     because data set is too small
 
                     Cannot compute OUTPUT(I) for I = 1,...,7
 
           IER = 2   Geometric mean results cannot be computed because
                     all x-values are equal
 
                     Cannot compute OUTPUT(I) for I = 1,2,3,4,7
 
           IER = 3   Cannot compute standard deviations of slope and intercept
                     estimates by geometric mean technique or correlation
                     coefficient because all y-values are equal
 
                     Cannot comput OUTPUT(I) for I = 3, 4, 7
 
    EXAMPLE
    -------
 
    INPUT:
 
         N = 10
 
         I          X(I)          Y(I)
         1          0.0           5.9
         2          0.9           5.4
         3          1.8           4.4
         4          2.6           4.6
         5          3.3           3.5
         6          4.4           3.7
         7          5.2           2.8
         8          6.1           2.8
         9          6.5           2.4
        10          7.4           1.5
 
    CALL SEQUENCE:
 
    call rgm (n, x, y, output, ier)
 
    OUTPUT:
 
          IER = 0
 
          OUTPUT(1) = -0.5526
          OUTPUT(2) =  5.8108
          OUTPUT(3) =  0.0377
          OUTPUT(4) =  0.2465
          OUTPUT(5) =  3.8200
          OUTPUT(6) =  3.7000
          OUTPUT(7) = -0.9765
 
    See documentation of SLRPACK subroutines RYORK and RWILL for
       continuations of this example.
 
    PRECISION
    ---------
 
    All calculations are done in single precision.
 
    LANGUAGE
    --------
 
    The routine is coded in standard Fortran 77.
 
    OTHER SUBROUTINES USED
    ----------------------
 
    PORT  subroutine  R1MACH
 
    REFERENCES
    ----------
 
    Kermack, K. A. and Haldane, J. B. S. (1950).  Organic Correlation
       and Allometry.  Biometrika, 37, 30-41.
 
    Pearson, K. (1901).  On lines and planes of closest fit to systems
       of points in space.  Phil. Mag. (6), 2, 559.
 
    Riggs, D. S., Guarnieri, J. A., and Addelman, S.  (1978).  Fitting
       straight lines when both variables are subject to error.  Life
       Sciences, 22, 1305-1360.
 
    NBS CONTACT
    -----------
 
    Sally E. Howe
    Scientific Computing Division
 
    CONTRIBUTORS
    ------------
 
    Sally E. Howe
    Kathryn Rensenbrink
    Gregory S. Rhoads
    Scientific Computing Division
    Center for Applied Mathematics
    National Bureau of Standards
    Gaithersburg, MD  20899
 
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